Official Solution:If \(n = x^5*y^7\), where \(x\) and \(y\) are positive integers greater than 1, then how many positive divisors does \(n\) have? (1) \(x\) does not have a factor \(p\) such that \(1 < p < x\) and \(y\) does not have a factor \(q\) such that \(1 < q < y\). This statement implies that both \(x\) and \(y\) are primes (a prime number does not have a factor which is greater than 1 and less than itself, it has only two factors 1 and itself). Now, if \(x\) and \(y\) are different primes, then the number of factors of \(n\) will be \((5+1)(7+1)=48\) but if \(x=y\), then \(n = x^5*y^7=x^{12}\) and it will have 13 factors. Not sufficient.

(2) \(n\) has only two prime factors. If those primes are \(x\) and \(y\), then the number of factors of \(n\) will be \((5+1)(7+1)=48\) but if, say \(x=2*3=6\) and \(y=2\), then \(n = x^5*y^7=2^{12}*3^5\) and it will have \((12+1)(5+1)=78\) factors. Not sufficient.

(1)+(2) Since from (2) \(n\) has only two prime factors, then from (1) it follows that \(x\) and \(y\) are different primes so the number of factors of \(n\) will be \((5+1)(7+1)=48\). Sufficient.

Answer: C

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